The manufacturing of semiconductor devices typically begins with a wafer of single crystal silicon and consists of three basic processes: doping, depositing or growing of thin films, and film patterning. The thin films are typically less than one micrometer thick. The thin films may be deposited on the surface of the wafer by sputtering or evaporation, or they may be grown by placing the wafer in a reactive environment. These thin films may be conductive materials, such as aluminum, tungsten, or polysilicon, or insulating materials, such as silicon nitride or silicon dioxide. The conductive films form device gates or interconnects, and the insulating films provide electrical isolation.
The thin films applied in the course of semiconductor manufacturing nearly always exhibit some intrinsic stress. This stress may be attributed to the mismatch of the thermal expansion coefficients of the crystal lattices of the film and substrate wafer, as well as other mechanisms. If these stresses are not controlled, they can lead to failure of electronic devices by a variety of mechanisms, including film cracking, delamination, and void formation. To some extent, film stress may be controlled during deposition or growth by process variation or afterward by annealing. But in order to accurately control film stress, the stress must be accurately measured.
Over the years, a wide variety of methods for measuring stress in thin films has emerged. These methods include X-ray diffraction of the film, micro-Raman spectroscopy of the film, use of micro-mechanical structures printed into the film, membrane resonant frequency measurement, and pressure bulge inducement. A very common technique for determining thin film stress in the semiconductor industry is the wafer curvature technique.
In order to determine film stress from wafer curvature, the out-of-plane distortion of the wafer must be measured. This measurement may be accomplished by several methods, including X-ray diffraction of the wafer (rather than the film), micro-Raman spectroscopy applied to the wafer, interferometry, capacitance gauging, profilometry, and laser scanning. As an example, a typical laser scanning technique is illustrated in FIG. 1. A diode laser, 101, generates a laser beam that is passed through lens 102, and scans across the wafer with the film, 103. The beam reflects off the wafer and is reflected by mirror 104 to a photodetector 105. The angle at which the beam is reflected is sensed and recorded. The angle is related to the rotation of the wafer""s surface. The measurement is typically repeated across several diameters of the wafer. The apparatus shown in FIG. 1 is typically embodied in a xe2x80x9cstress and flatness gaugexe2x80x9d such as the model FSM 128 Stress and Flatness Gauge manufactured by Frontier Semiconductor Measurements, Inc., of San Jose, Calif. Such an instrument is typically connected to a computer system, which determines the stress and displays a graphical stress map or other type of graphical display illustrating stresses across the film.
The stress on a film stretched across a wafer is currently computed in the above-described and similar arrangements by the following equation, commonly known in the industry as xe2x80x9cStoney""s equationxe2x80x9d:                               ∂          2                ⁢        w                    ∂                  y          2                      =                                        ∂            2                    ⁢          w                          ∂                      x            2                              =                        6          ⁢                      σ            f                    ⁢                                    t              f                        ⁢                          (                              i                -                v                            )                                                Et          s          2                      ,
where: "sgr"ƒ=film stress
tƒ=film thickness
ts=substrate thickness
xcexd=Poisson""s ratio of substrate
E=elastic modulus of substrate
w=out-of-plane displacement.
Although using this equation to compute film stress provides useful, qualitative information about the stress, the quantitative accuracy of film stresses determined with Stoney""s equation is severely limited. Stoney""s equation was derived using the assumption that the film stress is constant across the wafer. This assumption is often invalid as film stress usually varies across the wafer. A new technique for determining film stress from out-of-plane distortion of a substrate is needed to take varying film stress into account, and thus improve the accuracy of film stress measurements.
The present invention provides a method for calculating film stress based on an inverse finite element analysis of the displaced substrate wafer. The method of the invention produces accurate results regardless of whether film stress is constant across the wafer. The method can be employed with any system that measures substrate curvature and is not limited to the laser scanning system previously described.
In one embodiment, a stress measurement for a film applied to a substrate having known dimensions and known structural properties is made by first measuring the surface displacement of the wafer, and then determining structural compliance based on an inverse finite element model of the substrate, and structural stress-load using the known dimensions and known structural properties of the substrate. A stress field is calculated using the structural compliance, the structural stress-load, and the surface displacement of the substrate. The stress measurement is output based on values in the stress field. The structural compliance can be determined by calculating a stiffness, inverting the stiffness matrix, and removing columns corresponding to unloaded degrees of freedom and rows corresponding to unprescribed displacements. The structural stress-load can be determined by calculating element-by-element stress-loads as matrixes and expanding them.
In other embodiments, iteration is added to the method to further improve accuracy in some situations. Iteration can be added to the method by calculating surface loads based on the structural compliance, iteratively correcting the surface loads to arrive at final surface loads, and using the surface loads to calculate the stress field. In this case, the final surface load is eventually determined by determining a residual surface load at each iteration.
Iteration can also be added to the method of the invention by iteratively correcting coefficients in the stress field. In this embodiment, an initial stress field is calculated as described above, and the stress field coefficients are iteratively corrected until a final stress field is determined.
In example embodiments of the invention, computer software is used to implement many aspects of the invention. The software can be stored on a medium. The medium can be magnetic, such as a diskette, tape, or fixed disk, or optical, such as a CD-ROM or DVD-ROM. The software can also be stored in a semiconductor device. Additionally, the software can be supplied via the Internet or some other type of network. A workstation or computer system that typically runs the software. In one embodiment, this computer system, also called a xe2x80x9cprogram execution systemxe2x80x9d is interfaced to a measurement system such as the Stress and Flatness Gauge previously described. The software in combination with the computer system and measurement system forms the means to execute the method of the invention.